A few days ago I found myself writing “his fingers plucked at the strings” in one of my stories.

Well, wait. That doesn’t look right.

I’ll get the same kind of weird cringe when I see phrases like “his eyes watched” or “his foot stepped”. 

Why is that, though?

1. It adds unnecessary words. In modern literature (where we’re not paid by the word like in Dickens’ day), brevity is wit. These days, it’s more in vogue not to use two words when one will do. And yes, paying attention to the style of the day does matter, though it’s probably the least important thing on this list.
2. It implies unnecessary distinction. Unless you’re following somebody’s movements with your eyes but not actually paying attention, there’s no reason to point out that your eyes were watching them– and even then, there’s a better, more clear way of saying that. In every one of those cases, the ambiguous “he” is performing all of those actions, rather than the disparate parts of his body performing them of their own volition. 
3. It distances the action from the actor. He’s not tapping on the glass, his fingers are. He’s not watching the woman cross the room, his eyes are. This is the same kind of mental distance often employed by liars, when they say “the report got done” rather than “I did the report”.   
4. It makes some readers cranky. I can’t describe how many blog posts I’ve seen in which people will put down a book over phrases like “his eyes followed her across the room”, with the justification that that character’s eyes aren’t literally rolling out of his sockets and following a woman around. Normally I roll my eyes at these people– if this isn’t a sci-fi/fantasy book (or one that’s sufficiently gory), we all know exactly what the author was trying to say. But knowingly irritating readers is a dangerous game. And while we’re at it:
5. It can be unnecessarily confusing. If you are writing in a genre where body parts occasionally move of their own volition, you may need to be more careful with your turns of phrase, especially in situations where your readers haven’t yet learned exactly what is and isn’t possible. 

Now let me add the obvious: there’s an exception to every rule. Maybe you’re writing about autonomous body parts, or an unreliable narrator, or you really need to distinguish which of his hands is doing what at any given time.

I consider this to be a very minor writing mistake, to the point where I personally cite only one reason for avoiding it:

1. It just looks weird. 

And sometimes, that’s the only reason you need.